Research Article | Open Access

# Hermite-Hadamard Type Inequalities for Superquadratic Functions via Fractional Integrals

**Academic Editor:**Praveen Agarwal

#### Abstract

We use basic properties of superquadratic functions to obtain some new Hermite-Hadamard type inequalities via Riemann-Liouville fractional integrals. For superquadratic functions which are also convex, we get refinements of existing results.

#### 1. Introduction

Let be a convex function and with ; then is known as the Hermite-Hadamard inequality.

This remarkable result is well known in the literature as the Hermite-Hadamard inequality. Recently, the generalizations, refinements, and improvements of the classical Hermite-Hadamard inequality have been the subject of intensive research.

*Definition 1 (see [1]). *A function is superquadratic provided that for all there exists a constant such that
for all .

Theorem 2 (see [1]). *The inequality
**
holds for all probability measures and all nonnegative, -integrable functions , if and only if is superquadratic.*

The discrete version of the above theorem is also used in the sequel.

Lemma 3 (see [2]). *Suppose that is superquadratic. Let , , and let , where and . Then
*

Nonnegative superquadratic functions are much better behaved as we see next.

Lemma 4 (see [1]). *Let be a superquadratic function with as in Definition 1. Then one gets the following:*(1)*;*(2)*if , then whenever is differentiable at ;*(3)*if , then is convex and .*

The Hermite-Hadamard inequalities for superquadratic functions are established by Banic et al. in [3].

Theorem 5 (see [3]). *Let be a superquadratic function and ; then
*

It is remarkable that Sarikaya et al. [4] proved the following interesting inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional integrals.

Theorem 6 (see [4]). *Let be a positive function with and . If is a convex function on , then the following inequalities for fractional integrals hold:
**
with .*

We remark that the symbols and denote the left-sided and right-sided Riemann-Liouville fractional integrals of the order with which are defined by respectively. Here, is the gamma function defined by .

Fractional integral operators are widely used to solve differential equations and integral equations. So a lot of work has been obtained on the theory and applications of fractional integral operators.

For more results concerning the fractional integral operators, we refer the reader to [5–10] and references cited therein.

In this paper, we establish some new Hermite-Hadamard type inequalities for superquadratic functions via Riemann-Liouville fractional integrals which refine the inequalities of (6) for superquadratic functions which are also convex.

#### 2. Main Results

From Lemma 3 for , we get for , , for the superquadratic function that hold, and therefore Let ; we get that Therefore, by replacing in (9) we get that

Theorem 7. *Let be a superquadratic integrable function on with . Then
**
with .*

*Proof. *By inequality (12), we get that
By the change of variable , we get
Therefore
We have completed the proof.

Corollary 8. *Putting in Theorem 7 gives
*

Let ; we get that Therefore, by replacing in (9) we get that

Theorem 9. *Let be a superquadratic integrable function on with . Then
**
with .*

*Proof. *By inequality (20), we get that
The proof is completed.

Corollary 10. *Choosing in Theorem 9, one has
*

Corollary 11. *Let be defined as in Theorem 7; one gets
*

Corollary 12. *Taking in Corollary 11, one obtains
*

#### 3. Conclusion

In this note, we obtain some new Hermite-Hadamard type inequalities for superquadratic functions via Riemann-Liouville fractional integrals. For superquadratic functions which are also convex, we get refinements of known results. The concept of superquadratic functions in several variables is introduced in [11]. An interesting topic is whether we can use the methods in this paper to establish the Hermite-Hadamard inequalities for superquadratic functions in several variables via Riemann-Liouville integrals.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work is supported by the Youth Project of Chongqing Three Gorges University of China (no. 13QN11).

#### References

- S. Abramovich, G. Jameson, and G. Sinnamon, “Refining Jensen’s inequality,”
*Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie*, vol. 47, no. 95, pp. 3–14, 2004. View at: Google Scholar | MathSciNet - S. Abramovich, J. Barić, and J. Pečarić, “Fejer and Hermite-Hadamard type inequalities for superquadratic functions,”
*Journal of Mathematical Analysis and Applications*, vol. 344, no. 2, pp. 1048–1056, 2008. View at: Publisher Site | Google Scholar | MathSciNet - S. Banic, J. Pečari, and S. Varošanec, “Superquadratic functions and refinements of some classical inequalities,”
*Journal of the Korean Mathematical Society*, vol. 45, no. 2, pp. 513–525, 2008. View at: Publisher Site | Google Scholar | MathSciNet - M. Z. Sarikaya, E. Set, H. Yaldiz, and N. Başak, “Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities,”
*Mathematical and Computer Modelling*, vol. 57, no. 9-10, pp. 2403–2407, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH - P. Agarwal, “Certain properties of the generalized Gauss hypergeometric functions,”
*Applied Mathematics & Information Sciences*, vol. 8, no. 5, pp. 2315–2320, 2014. View at: Publisher Site | Google Scholar | MathSciNet - J. Choi and P. Agarwal, “Certain fractional integral inequalities involving hypergeometric operators,”
*East Asian Mathematical Journal*, vol. 30, no. 3, pp. 283–291, 2014. View at: Google Scholar - J. Choi and P. Agarwal, “Certain new pathway type fractional integral inequalities,”
*Honam Mathematical Journal*, vol. 36, no. 2, pp. 437–447, 2014. View at: Google Scholar - J. Choi and P. Agarwal, “Some new Saigo type fractional integral inequalities and their
*q*-analogues,”*Abstract and Applied Analysis*, vol. 2014, Article ID 579260, 11 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - D. Baleanu and P. Agarwal, “On generalized fractional integral operators and the generalized Gauss hypergeometric functions,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 630840, 5 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - D. Baleanu and P. Agarwal, “Certain inequalities involving the fractional $q$-integral operators,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 371274, 10 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - S. Abramovich, S. Banić, and M. Matić, “Superquadratic functions in several variables,”
*Journal of Mathematical Analysis and Applications*, vol. 327, no. 2, pp. 1444–1460, 2007. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2014 Guangzhou Li and Feixiang Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.